Page 116 - Advanced Linear Algebra
P. 116
100 Advanced Linear Algebra
theorem implies that
b 8 b- b 8 b (( b ~8 ² 8 ³ b -
This shows that
2 8 3 8
dim - ²- ³ dim - ²- ³
and completes the proof.
Reflexivity
=
If is a vector space, then so is the dual space = i and so we may form the
)
(
double algebraic dual space = ii , which consists of all linear functionals
**
i ¢= ¦ -. In other words, an element of = is a linear functional that
assigns a scalar to each linear functional on .
=
With this firmly in mind, there is one rather obvious way to obtain an element of
= ii # . Namely, if = , consider the map # ¢ = i ¦ - defined by
#² ³ ~ ²#³
#
which sends the linear functional to the scalar ² # ³ . The map is called
i
evaluation at #. To see that # = ii , if Á = and Á - , then
#² b ³ ~ ² b ³²#³ ~ ²#³ b ²#³ ~ #² ³ b #² ³
and so is indeed linear.
#
We can now define a map ¢= ¦ = ii by
#~#
This is called the canonical map or the natural map) from to = ii . This
(
=
map is injective and hence in the finite-dimensional case, it is also surjective.
Theorem 3.13 The canonical map ii defined by ¢= ¦ = # ~ # , where is
#
evaluation at , is a monomorphism. If is finite-dimensional, then is an
#
=
isomorphism.
Proof. The map is linear since
" b #² ³ ~ ² " b #³ ~ ²"³ b ²#³ ~ ² " b #³² ³
for all = i . To determine the kernel of , observe that
#~ ¬#~
¬#² ³~ for all = i
¬ ²#³~ for all = i
¬# ~
by Theorem 3.10 and so ker² ³ ~ ¸ ¹ . In the finite-dimensional case, since
